How to smooth area with sharp edges?
The Subdivision surface object modifier

There are two things that I've identified in your screenshots that need to be improved. One is a fix, and one is the modifier.

Mesh Topology

Tab into Edit Mode and key A once or twice until everything is selected. Key Alt + J to convert Tris to Quads (triangular faces to quad faces). You can also key Ctrl + F to open the Faces menu and manually select Tris to Quads. Lastly, you can go 3D Viewport > bottom menu > Mesh > Faces > Tris to Quads.

Ctrl + F menu:



Mesh > Faces menu:



Subdivision surface

The Math

This will use the Catmull-Clark algorithm to create new faces between two imaginary points in the center of each current face.
From Wikipedia:


  Catmull–Clark surfaces are defined recursively, using the following refinement scheme:
  
  Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points.
  
  
  For each face, add a face point.
  
  
  Set each face point to be the average of all original points for the respective face.
  
  For each edge, add an edge point.
  
  
  Set each edge point to be the average of the two neighbouring face points and its two original endpoints.
  For each face point, add an edge for every edge of the face, connecting the face point to each edge point for the face.
  
  For each original point P, take the average F of all n (recently created) face points for faces touching P, and take the average R of all n edge midpoints for (original) edges touching P, where each edge midpoint is the average of its two endpoint vertices (not to be confused with new "edge points" above). Move each original point to the point:
  
  
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
  
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This is the barycenter of P, R and F with respective weights (n − 3), 2 and 1.
  
  
  Connect each new vertex point to the new edge points of all original edges incident on the original vertex.
  Define new faces as enclosed by edges.
  


The Modifier

Go to the Properties panel > Object modifiers tab and add a Subdivision surface modifier. The two values you should care about are the View: and the Render: values. The View: value determines the number of subsurf levels, or iterations, displayed in the 3D Viewport. The Render: value determines the number of subsurf levels in a rendered image or animation. The Render: value should generally be higher.

The Subdivision surface modifier:



Subsurf levels 0, 1, 2, and 3 on a Suzanne: